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圆周角定理动态演示

圆周角定理动态演示
几何画板及绘图内容加载中，请稍等……
2018-03-01 09:04
圆角三角函数几何作图滑块动画
所属专题 Circles (Theorems)
①本范例动态演示圆周角定理的几何意义，通过旋转和交点变化直观展示圆周角与所对圆心角的关系；②涉及核心数学概念包括圆周角定理、圆心角、旋转、相似三角形等；③作图要点运用了Circle（圆）、Point（点）、CircleArc（圆弧）、Angle（角）、Rotate（旋转）、Ray（射线）、Dilate（缩放）、Segment（线段）、Intersect（交点）、CircleSector（扇形）、Vector（向量）、Translate（平移）、Line（直线）等GeoGebra命令和对象；④适合高中数学教学，用于加深学生对圆周角定理的理解和应用。

GeoGebra 命令及参数

  1. Circle(A, B) → c
  2. Point(c) → C
  3. Point(c) → D
  4. CircleArc(A, D, C) → d
  5. Point(d) → E
  6. Point(d) → F
  7. Point(d) → G
  8. Point(d) → H
  9. Point(d) → I
  10. CircleArc(A, C, D) → e
  11. Angle(C, A, D) → α
  12. Rotate(C, α / 2, A) → C'
  13. Angle(C, A, C') → β
  14. Ray(A, C') → h
  15. Dilate(A, If[0 < a ≤ 1, 1 - a, 0], C) → A'
  16. Dilate(A, If[0 < a ≤ 1, 1 - a, 0], D) → A'_1
  17. Segment(A, A') → f
  18. Segment(A, A'_1) → g
  19. Segment(A, B) → i
  20. Segment(A, C) → j
  21. Circle(A, (0.3 * i)) → k
  22. Intersect(k, j) → K,
  23. Dilate(E, If[1 < a ≤ 2, 2 - a, 0], C) → E'
  24. Dilate(E, If[1 < a ≤ 2, 2 - a, 0], D) → E'_1
  25. Dilate(F, If[1 < a ≤ 2, 2 - a, 0], D) → F'
  26. Dilate(G, If[1 < a ≤ 2, 2 - a, 0], D) → G'
  27. Dilate(H, If[1 < a ≤ 2, 2 - a, 0], D) → H'
  28. Dilate(I, If[1 < a ≤ 2, 2 - a, 0], D) → I'
  29. Dilate(F, If[1 < a ≤ 2, 2 - a, 0], C) → F'_1
  30. Dilate(G, If[1 < a ≤ 2, 2 - a, 0], C) → G'_1
  31. Dilate(H, If[1 < a ≤ 2, 2 - a, 0], C) → H'_1
  32. Dilate(I, If[1 < a ≤ 2, 2 - a, 0], C) → I'_1
  33. Segment(E, E') → l
  34. Segment(E, E'_1) → m
  35. Segment(F, F'_1) → n
  36. Segment(F, F') → p
  37. Segment(G, G'_1) → q
  38. Segment(G, G') → r
  39. Segment(H, H'_1) → s
  40. Segment(H, H') → t
  41. Segment(I, I'_1) → b
  42. Segment(I, I') → f_1
  43. Dilate(A, If[0 < a ≤ 1, 1 - a, 0], C') → A'_2
  44. Segment(A, A'_2) → i_1
  45. Rotate(K, If[1 < a ≤ 2, ((a - 1) * α), α], A) → K'
  46. CircleSector(A, K, K') → c_1
  47. CircleSector(A, D, C) → d_1
  48. CircleSector(A, C, D) → e_1
  49. Rotate(K, If[0 < a ≤ 1, (a * α / 2), α / 2], A) → K'_1
  50. CircleSector(A, K, K'_1) → k_1
  51. Ray(E, C) → j_1
  52. Ray(F, C) → l_1
  53. Ray(G, C) → m_1
  54. Ray(H, C) → n_1
  55. Ray(I, C) → p_1
  56. Circle(E, (0.3 * i)) → r_1
  57. Circle(F, (0.3 * i)) → s_1
  58. Circle(G, (0.3 * i)) → t_1
  59. Circle(H, (0.3 * i)) → c_2
  60. Circle(I, (0.3 * i)) → d_2
  61. Intersect(r_1, j_1) → L,
  62. Intersect(n, l_1) → M
  63. Intersect(s_1, l_1) → N,
  64. Intersect(t_1, m_1) → O,
  65. Intersect(c_2, n_1) → P,
  66. Intersect(d_2, p_1) → Q,
  67. Rotate(Q, If[1 < a ≤ 2, ((a - 1) * α / 2), α / 2], I) → Q'
  68. CircleSector(I, Q, Q') → f_2
  69. Rotate(k_1, If[2 < a ≤ 3, ((a - 2) * α / 2), α / 2], A) → g_2
  70. Dilate(I, If[3 < a ≤ 4, 4 - a, 0], F) → I'_2
  71. Vector(I, I'_2) → u
  72. Translate(f_2, u) → h_2
  73. Line(I, n) → a_1
  74. Point(a_1) → S
  75. Angle(Q, I, S) → γ
  76. Rotate(h_2, If[3 < a ≤ 4, ((-(a - 3)) * ((2 * π) - γ)), γ - (2 * π)], I'_2) → k_2
  77. Dilate(A, If[3.5 < a ≤ 4.5, 4.5 - a, 0], I) → A'_3
  78. Vector(A, A'_3) → v
  79. Translate(g_2, v) → p_2
  80. Rotate(K, If[3.5 < a ≤ 4.5, ((a - 3.5) * α / 2), α / 2], A) → K'_2
  81. CircleSector(A, K, K'_2) → q_2
  82. Line(I, i_1) → b_1
  83. Point(b_1) → T
  84. Angle(Q, I, T) → δ
  85. Rotate(p_2, If[3.5 < a ≤ 4.5, ((a - 3.5) * ((2 * π) - δ)), (2 * π) - δ], A'_3) → r_2
  86. Dilate(F, If[4.5 < a ≤ 5.5, 5.5 - a, 0], G) → F'_2
  87. Vector(F, F'_2) → w
  88. Translate(k_2, w) → s_2
  89. Line(G, n) → i_2
  90. Point(i_2) → U
  91. Angle(O, G, U) → ε
  92. Rotate(s_2, If[4.5 < a ≤ 5.5, ((-(a - 4.5)) * ε), (-ε)], F'_2) → t_2
  93. Rotate(L, If[3 < a ≤ 4, ((a - 3) * α / 2), α / 2], E) → L'
  94. CircleSector(E, L, L') → c_3
  95. Dilate(E, If[5 < a ≤ 6, 6 - a, 0], H) → E'_2
  96. Vector(E, E'_2) → u_1
  97. Translate(c_3, u_1) → d_3
  98. Line(H, l) → j_2
  99. Point(j_2) → V
  100. Angle(P, H, V) → ζ
  101. Rotate(d_3, If[5 < a ≤ 6, ((a - 5) * ((2 * π) - ζ)), (2 * π) - ζ], E'_2) → e_3
  102. Dilate(A, If[5.5 < a ≤ 6.5, 6.5 - a, 0], E) → A'_4
  103. Vector(A, A'_4) → v_1
  104. Translate(q_2, v_1) → f_3
  105. Line(E, f) → l_2
  106. Point(l_2) → W
  107. Angle(W, E, C) → η
  108. Rotate(f_3, If[5.5 < a ≤ 6.5, ((a - 5.5) * η), η], A'_4) → g_3
  109. Dilate(I, If[6 < a ≤ 7, 7 - a, 0], F) → I'_3
  110. Vector(I, I'_3) → w_1
  111. Translate(r_2, w_1) → g_1
  112. Line(F, b) → h_1
  113. Point(h_1) → J
  114. Angle(J, F, N) → θ
  115. Rotate(g_1, If[6 < a ≤ 7, ((-(a - 6)) * ((2 * π) - θ)), θ - (2 * π)], I'_3) → q_1
  116. Rotate(K, If[6.5 < a ≤ 7.5, ((a - 6.5) * β), β], A) → K'_3
  117. CircleSector(A, K, K'_3) → e_2
  118. Dilate(H, If[7 < a ≤ 8, 8 - a, 0], I) → H'_2
  119. Vector(H, H'_2) → u_2
  120. Translate(e_3, u_2) → h_3
  121. Line(I, s) → m_2
  122. Point(m_2) → R
  123. Angle(R, I, Q) → κ
  124. Rotate(h_3, If[8 < a ≤ 9, ((a - 8) * κ), κ], I) → k_3
  125. Dilate(A, If[8 < a ≤ 9, 9 - a, 0], H) → A'_5
  126. Vector(A, A'_5) → v_2
  127. Translate(e_2, v_2) → p_3
  128. Line(H, f) → n_2
  129. Point(n_2) → Z
  130. Angle(Z, H, P) → λ
  131. Rotate(p_3, If[8 < a ≤ 9, ((a - 8) * λ), λ], A'_5) → q_3